VREL‑A — Vertical Resonance Extraction Lens (Acoustic Variant)

A SARG lens for extracting harmonic invariants from any substrate.

VREL‑A is the acoustic variant of VREL. Where VREL reads a substrate through mirror‑axis symmetry, VREL‑A reads it through oscillatory structure — frequency, phase, rhythm, and overtone relationships.

VREL‑A does not require the substrate to be audible. Any substrate that carries periodic, quasi‑periodic, or oscillatory behavior can be read acoustically: heartbeats, tidal cycles, crystal vibrations, orbital periods, neural firing patterns, stellar pulsations.

VREL‑A is substrate‑agnostic: it operates identically whether the input is a birdsong, a seismic trace, a circadian rhythm, or a pulsar envelope.

1. What VREL‑A Extracts#

VREL‑A applies harmonic‑family analysis to a substrate and returns three invariant classes:

1.1 Harmonic Invariants#

Features that persist across the substrate's frequency spectrum — the tones that remain stable when the signal is decomposed into its component frequencies.

• In acoustic substrates: fundamental frequency, dominant overtones, formant peaks
• In biological substrates: heartbeat base rate, respiratory cadence, circadian period
• In crystalline substrates: lattice vibration modes (phonon frequencies)
• In cosmological substrates: orbital resonance ratios, pulsar spin frequencies
• In linguistic substrates: vowel formants, prosodic pitch contours

1.2 Rhythmic Invariants#

Features that persist across the substrate's temporal structure — the patterns that remain stable when timing is analyzed independently of pitch.

• In acoustic substrates: beat patterns, rhythmic groupings, meter signatures
• In biological substrates: gait cycles, peristaltic rhythms, firing‑burst intervals
• In geological substrates: tidal periodicities, eruption intervals, seismic recurrence
• In cosmological substrates: orbital periods, rotation‑revolution ratios
• In symbolic substrates: repetition intervals, sequence cadences

1.3 Phase‑Coherent Invariants#

Features that persist when both frequency and rhythm are analyzed together — elements whose phase relationships remain locked across transformations.

• In acoustic substrates: consonant intervals, stable overtone‑to‑fundamental ratios
• In biological substrates: synchronized oscillators (e.g., firefly flash sync, neural coherence)
• In crystalline substrates: phonon modes that maintain phase across grain boundaries
• In cosmological substrates: orbital resonance locks (e.g., Jupiter–Io 4:1)
• In lostational substrates: resonance shells that maintain coherence across the visible–inverted boundary

Phase‑coherent invariants are the most structurally stable elements in any oscillatory substrate. They are the strongest candidates for resonance anchor mapping.

2. How VREL‑A Behaves Across Substrates#

VREL‑A adapts its extraction logic to the substrate's native oscillatory structure. The lens itself does not change — the interpretation of "oscillation" changes.

The same lens produces the same invariant types (harmonic, rhythmic, phase‑coherent) regardless of domain. Only the substrate determines what those invariants are.

3. Invariants Produced by VREL‑A#

Every VREL‑A extraction produces a structured invariant set:

"invariants": {
  "harmonic": ["f₀", "2f₀", "3f₀", "5f₀"],
  "rhythmic": ["4/4 pulse", "dotted‑pair grouping", "hemiola"],
  "phase_coherent": ["f₀–2f₀ lock", "3f₀–5f₀ lock"]
}

Properties of VREL‑A invariants#

• Harmonic invariants describe what oscillates — the frequency content.
• Rhythmic invariants describe when it oscillates — the temporal patterning.
• Phase‑coherent invariants are always the intersection of harmonic and rhythmic — elements locked in both dimensions simultaneously.
• The phase‑coherent set contains the substrate's most resonance‑stable elements.
• An empty phase‑coherent set indicates low oscillatory coherence — the substrate may be aperiodic, chaotic, or require a different lens resolution.

Coherence stability rule#

If an element appears in the phase‑coherent set, it is guaranteed to persist under any single‑axis transformation (frequency shift or tempo change alone). Phase‑coherent invariants are the anchoring layer of the substrate.

4. Resonance Anchor Mapping#

VREL‑A invariants map to the same four universal resonance anchors as VREL, but through an acoustic interpretation:

Mapping logic#

VREL‑A maps each phase‑coherent invariant to the anchor whose acoustic root it most closely resembles:

"resonance_mapping": {
  "universal_anchors": [
    { "form": "f₀", "mapped_to": "line" },
    { "form": "f₀–2f₀ lock", "mapped_to": "circle" },
    { "form": "3f₀–5f₀ beating", "mapped_to": "cross" },
    { "form": "onset transient", "mapped_to": "dot" }
  ]
}

• Pure carrier forms (fundamental, drone, sustained tone) → map to line
• Pure cyclic forms (standing wave, looped oscillation, stable orbit) → map to circle
• Pure interference forms (beating, consonance nodes, harmonic crossing) → map to cross
• Pure impulse forms (onset, attack, percussive seed) → map to dot
• Hybrid forms (e.g., a pulsed drone) → map to compound anchors (e.g., dot+line_hybrid)

Forms that do not cleanly map to any anchor are classified as anchorless and routed to the SARG error taxonomy (see error/).

Confidence scoring#

Each mapping carries an optional confidence value (0.0–1.0):

{ "form": "f₀", "mapped_to": "line", "confidence": 1.0 }
{ "form": "3f₀–5f₀ beating", "mapped_to": "cross", "confidence": 0.7 }

Confidence reflects how unambiguously the invariant maps to a single anchor. Phase‑coherent invariants with high confidence are the substrate's resonance core.

5. VREL‑A in the SARG Object#

Every SARG object that uses VREL‑A includes a lens block:

"lens": {
  "type": "VREL‑A",
  "variant": "standard",
  "version": "1.0.0",
  "notes": "Harmonic‑family resonance extraction for oscillatory substrates."
}

• type — always VREL‑A for this lens
• variant — standard for the base lens; future variants may include spectral, rhythmic‑only, or phase‑locked
• version — semantic version for tracking lens evolution
• notes — any special considerations for this extraction

6. Relationship to VREL#

VREL and VREL‑A are complementary, not competing.

Both lenses produce the same invariant types (a primary set, a secondary set, and an intersection set) and map to the same universal anchors (● ○ × |).

On lostational substrates, both lenses can be applied simultaneously: VREL reads the dimensional shape; VREL‑A reads the dimensional pulse. Together they produce a full resonance fingerprint — structure + oscillation.

7. Relationship to Other Files#

• lens_overview.md — what a lens is, how lenses fit into SARG
• VREL.md — the spatial (mirror‑axis) parent lens
• invariants/invariant_types.md — detailed invariant classification
• resonance/resonance_mapping.md — full anchor mapping logic
• resonance/resonance_families.md — how invariants group into families
• examples/ — working SARG objects using VREL‑A
• error/ — what happens when VREL‑A extraction fails or produces ambiguous results

What's in the file#

The file mirrors VREL.md's architecture so the two read as a matched pair — spatial lens and acoustic lens, same grammar, same anchors, complementary perspectives.